A generalization of Lévy's concentration-variance inequality

Summary Sharp lower bounds are found for the concentration of a probability distribution as a function of the expectation of any given convex symmetric function . In the case (x)=(x-c)², wherec is the expected value of the distribution, these bounds yield the classical concentration-variance inequality of Lévy. An analogous sharp inequality is obtained in a similar linear search setting, where a sharp lower bound for the concentration is found as a function of the maximum probability swept out from a fixed starting point by a path of given length.Research partially supported by NSF Grant SES-88-21999Research partially supported by NSF Grants DMS-87-01691 and DMS-89-01267 and a Fulbright Research Grant     

A generalization of the Pólya–Vinogradov inequality

In this paper we consider an approach of Dobrowolski and Williams which leads to a generalization of the Pólya–Vinogradov inequality. We show how the Dobrowolski–Williams approach is related to the classical proof of Pólya–Vinogradov using Fourier analysis. Our results improve upon the earlier work of Bachman and Rachakonda (Ramanujan J. 5:65–71, 2001). In passing, we also obtain sharper explicit versions of the Pólya–Vinogradov inequality.     

A Cramér–Rao inequality for non-differentiable models

We compute a variance lower bound for unbiased estimators in statistical models. The construction of the bound is related to the original Cramér–Rao bound, although it does not require the differentiability of the model. Moreover, we show our efficiency bound to be always greater than the Cramér–Rao bound in smooth models, thus providing a sharper result.     

A generalization of a basic inequality of Telyakovski?

Two classes of numerical sequences are defined by means certain properties of the differences of terms of positive sequences, and their relationship to some newly defined classes and the well-known Sidon-Telyakovski? class is analyzed. It is also verified that if a sequence belongs to the newly defined wider class, then with this sequence in place of the sequence {1/k} an essential inequality established by Telyakovski? can be generalized notably. This new result and the previous generalizations of the original Telyakovski???s theorem are incomparable.     

A functional generalization of the Cauchy–Schwarz inequality and some subclasses

In this work, a functional generalization of the Cauchy–Schwarz inequality is presented for both discrete and continuous cases and some of its subclasses are then introduced. It is also shown that many well-known inequalities related to the Cauchy–Schwarz inequality are special cases of the inequality presented.     

A generalization of the Chung-Erdös inequality for the probability of a union of events

A generalization of the Chung-Erdös inequality for the probability of a union of arbitrary events is proved using some lower bounds for tail probabilities. We present a lower bound for the probability of appearance of at least m events from a set of events A₁,..., A_n, where 1 ≤ m ≤ n. Bibliography: 6 titles.     

On a generalization of Aczél’s inequality

In this paper, we prove a generalization of Aczél’s inequality. The obtained inequalities extend some results established recently. We also give some comments on a recent result concerning the refinements of the generalized Aczél–Popoviciu’s inequality.     

A remark on the series of H. Cramér

One establishes some asymptotic representations for an analogue of H. Cramér's series, with the aid of which one describes the asymptotics of the probabilities of large deviations of the norm of the sum of independent, identically distributed random variables in a Hilbert space (L. V. Osipov, Teor. Veroyatn. Primen.,23, 510–526).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 181–185, 1980.     

A generalization of Gauss-Kuzmin-Lévy theorem

We prove a generalized Gauss-Kuzmin-Lévy theorem for the generalized Gauss transformation

$T_p\left(x\right)=\left\{\frac{p}{x}\right\}.$

In addition, we give an estimate for the constant that appears in the theorem.     

A generalization of the Pólya–Szegö inequality for one-dimensional functionals

We consider the Pólya–Szegö type weighted inequality. We prove this inequality for monotone rearrangement and for Steiner’s symmetrization.     

A generalization of Chebyshev’s inequality for Hilbert-space-valued random elements

We obtain a new generalization of Chebyshev’s inequality for random vectors. Then we extend this result to random elements taking values in a separable Hilbert space.     

An Unusual Application of Cramér-Rao Inequality to Prove the Attainable Lower Bound for a Ratio of Complicated Gamma Functions

Methodology and Computing in Applied Probability - A specific function f(r) involving a ratio of complicated gamma functions depending upon a real variable r(>?0) is handled. Details...     

Optimal Estimation and Cramér-Rao Bounds for Partial Non-Gaussian State Space Models

Partial non-Gaussian state-space models include many models of interest while keeping a convenient analytical structure. In this paper, two problems related to partial non-Gaussian models are addressed. First, we present an efficient sequential Monte Carlo method to perform Bayesian inference. Second, we derive simple recursions to compute posterior Cramér-Rao bounds (PCRB). An application to jump Markov linear systems (JMLS) is given.     

A variational formula on the Cramér function of series of independent random variables

In (Zajkowski, Positivity 19:529–537, 2015) it has been proved some variational formula on the Legendre–Fenchel transform of the cumulant generating function (the Cramér function) of Rademacher series with coefficients in the space \(\ell ^1\). In this paper we show a generalization of this formula to series of a larger class of any independent random variables with coefficients that belong to the space \(\ell ^2\).     

A generalization of the Shapley–Ichiishi result

The Shapley–Ichiishi result states that a game is convex if and only if the convex hull of marginal vectors equals the core. In this paper, we generalize this result by distinguishing equivalence classes of balanced games that share the same core structure. We then associate a system of linear inequalities with each equivalence class, and we show that the system defines the class. Application of this general theorem to the class of convex games yields an alternative proof of the Shapley–Ichiishi result. Other applications range from computation of stable sets in non-cooperative game theory to determination of classes of TU games on which the core correspondence is additive (even linear). For the case of convex games we prove that the theorem provides the minimal defining system of linear inequalities. An example shows that this is not necessarily true for other equivalence classes of balanced games.     

A generalization of the Dedekind–MacNeille completion

In this paper we introduce the notion of \(Z_{\delta }\)-continuity as a generalization of precontinuity, complete continuity and \(s_{2}\)-continuity, where Z is a subset selection. And for each poset P, a closure space \(Z^{c}_{\delta }(P)\) arises naturally. For any subset system Z, we define a new type of completion, called \(Z_{\delta }\)-completion, extending each poset P to a Z-complete poset. The main results are: (1) if a subset system Z is subset-hereditary, then \(cl_{Z}(\Psi (P))\), the Z-closure of all principal ideals \(\Psi (P)\) of poset P in \(Z^{c}_{\delta }(P)\), is a \(Z_{\delta }\)-completion of P and \(Z^{c}_{\delta }(P) \cong Z^{c}_{\delta }(cl_{Z}(\Psi (P)))\); (2) let Z be an HUL-system and P a \(Z_{\delta }\)-continuous poset, then the \(Z_{\delta }\)-completion of P is also \(Z_{\delta }\)-continuous, and a Z-complete poset L is a \(Z_{\delta }\)-completion of P iff P is an embedded \(Z_{\delta }\)-basis of L; (3) the Dedekind–MacNeille completion is a special case of the \(Z_{\delta }\)-completion.